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Hypernumbers and Other Exotic Stuff HYPERNUMBERS AND OTHER EXOTIC STUFF Photo by Mateusz Dach (1) MORE ON THE "ARITHMETICAL" SIDE Tropical Arithmetics Introduction to Tropical Geometry - Diane Maclagan and Bernd Sturmfels http://www.cs.technion.ac.il/~janos/COURSES/238900-13/Tropical/MaclaganSturmfels.pdf https://en.wikipedia.org/wiki/Min-plus_matrix_multiplication https://en.m.wikipedia.org/wiki/Tropical_geometry#Algebra_background https://en.wikipedia.org/wiki/Amoeba_%28mathematics%29 https://www.youtube.com/watch?v=1_ZfvQ3o1Ac (friendly introduction) https://en.wikipedia.org/wiki/Log_semiring https://en.wikipedia.org/wiki/LogSumExp Tight spans, Isbell completions and semi-tropical modules - Simon Willerton https://arxiv.org/pdf/1302.4370.pdf (one half of the tropical semiring) Hyperfields for Tropical Geometry I. Hyperfields and dequantization - Oleg Viro https://arxiv.org/pdf/1006.3034.pdf (see section "6. Tropical addition of complex numbers") Supertropical quadratic forms II: Tropical trigonometry and applications - Zur Izhakian, Manfred Knebusch and Louis Rowen - https://www.researchgate.net/publication/ 326630264_Supertropical_Quadratic_forms_II_Tropical_Trigonometry_and_Applications Tropical geometry to analyse demand - Elizabeth Baldwin and Paul Klemperer http://elizabeth-baldwin.me.uk/papers/baldwin_klemperer_2014_tropical.pdf International Trade Theory and Exotic Algebras - Yoshinori Shiozawa https://link.springer.com/article/10.1007/s40844-015-0012-3 Arborescent numbers: higher arithmetic operations and division trees - Henryk Trappmann http://eretrandre.org/rb/files/Trappmann2007_81.pdf Tetration Reference - Henryk Trappman and Andrew Robbins https://math.eretrandre.org/tetrationforum/attachment.php?aid=387 Open problems in tetration https://math.eretrandre.org/tetrationforum/showthread.php?tid=162 Applications of tetration https://math.stackexchange.com/questions/199862/what-is-the-geometric-physical-or-other-meaning- of-the-tetration The family of arithmetics of Ruggero Maria Santilli http://www.santilli-foundation.org/docs/10.11648.j.ajmp.s.2015040501.14.pdf Isodual Theory of Antimatter with applications to Antigravity, Grand Unification and Cosmology https://www.amazon.com/Isodual-Theory-Antimatter-applications-Antigravity/dp/1402045174 (book) Studies on Santilli’s Isonumber Theory - Arun S. Muktibodh http://www.santilli-foundation.org/docs/pdf2.pdf Elements of Hadronic Mechanics III Experimental verifications - R.M.Santilli http://www.santilli-foundation.org/docs/elements-hadronic-mechanics-iii.compressed.pdf Initiating Santilli's Iso-Mathematics to Triplex Number... - Nathan O. Schmidt and Reza Katebi http://vixra.org/pdf/1308.0051v2.pdf http://thunder-energies.com/ && http://www.santilli-foundation.org The Development of Hyper-Dual Numbers for Exact Second-Derivative Calculations Jeffrey A. Fike and Juan J. Alonso http://adl.stanford.edu/hyperdual/Fike_AIAA-2011-886_slides.pdf Saturation Arithmetics https://en.wikipedia.org/wiki/Saturation_arithmetic Symmetric level index - https://en.wikipedia.org/wiki/Symmetric_level-index_arithmetic A Hybrid Number Representation Scheme Based on Symmetric Level-Index Arithmetic Xunyang Shen and Peter R. Turner - https://www.researchgate.net/publication/ 221142816_A_Hybrid_Number_Representation_Scheme_Based_on_Symmetric_Level- Index_Arithmetic ZEA A zero-free exact arithmetic - Dominique Michelucci and Jean-Michel Moreau https://www.researchgate.net/publication/220991026_ZEA_-_A_zero-free_exact_arithmetic Algebra of screws https://en.wikipedia.org/wiki/Screw_theory#Algebra_of_screws On quantum state of numbers - Bernard Le Stum & Adolfo Quirós https://arxiv.org/pdf/1310.8143.pdf Half-exponential function - https://en.wikipedia.org/wiki/Half-exponential_function Matrix exponential - https://en.wikipedia.org/wiki/Matrix_exponential Baker–Campbell–Hausdorff formula https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula N-ary group https://en.wikipedia.org/wiki/N-ary_group Circuits over sets of natural numbers https://en.wikipedia.org/wiki/Circuits_over_sets_of_natural_numbers The complexity of circuit evaluation over the natural numbers - Pierre McKenzie and Klaus Wagner http://www.iro.umontreal.ca/~mckenzie/Dagstuhl02.pdf The Unwinding Number - Robert M. Corless and David J. Jeffrey https://faculty.e-ce.uth.gr/akritas/CE102/p28-corless.pdf Generalized distributive law https://en.wikipedia.org/wiki/Generalized_distributive_law A Noncommutative Version of the Natural Numbers - Tyler Foster https://arxiv.org/pdf/1003.2081.pdf A new number system: Remainder numbers https://math.stackexchange.com/questions/2415896/a-new-number-system-remainder-numbers Generalization of the unit interval - William M. Cornette https://projecteuclid.org/download/pdf_1/euclid.pjm/1102818012 Numeristics - Kevin Carmody - https://kevincarmody.com/math/numeristics.pdf Construction, properties and applications of finite neofield - Anthony Donald Keedwell https://dml.cz/bitstream/handle/10338.dmlcz/119164/CommentatMathUnivCarolRetro_41-2000- 2_8.pdf Quantity Calculus - https://en.wikipedia.org/wiki/Quantity_calculus http://ingvar.web03.cefit.se/wp-content/uploads/2016/02/physics6.pdf Metrological Thinking Needs the Notions of Parametric Quantities, Units, and Dimensions Ingvar Johansson Solving Cubic Equations with Curly Roots - Dan Kalman and Maurice Burke https://www.jstor.org/stable/10.5951/mathteacher.108.5.0392?seq=1 The eightfold path to nonstandard analysis - Vieri Benci, Mauro Di Nasso and Marco Forti https://www.researchgate.net/profile/Vieri_Benci/publication/ 228753190_The_eightfold_path_to_nonstandard_analysis/links/0deec52e248b66edc1000000/The- eightfold-path-to-nonstandard-analysis.pdf Radical of an integer - https://en.wikipedia.org/wiki/Radical_of_an_integer Integer square root - https://en.wikipedia.org/wiki/Integer_square_root Quadratic residue - https://en.wikipedia.org/wiki/Quadratic_residue Hypernumbers and Extrafunctions: Extending the Classical Calculus - Mark Burgin https://www.amazon.com/Hypernumbers-Extrafunctions-Extending-SpringerBriefs-Mathematics/dp/ 1441998748 Ordinal number - https://en.wikipedia.org/wiki/Ordinal_number Mex - https://en.wikipedia.org/wiki/Mex_(mathematics) Parallel - https://en.wikipedia.org/wiki/Parallel_(operator) Alternative models of the real number line in physics - D. K. Ross https://link.springer.com/article/10.1007/BF02213428 Can There Be an Alternative Mathematics, Really? - Jean Paul Van Bendegen https://link.springer.com/chapter/10.1007%2F0-387-24270-8_30 How Much Mathematics Is “Hardwired” If Any at All - Rafael Núñez https://cogsci.ucsd.edu/~nunez/COGS152_Readings/Nunez_ch3_MN.pdf Fractions in transrational arithmetic - Jan A. Bregstra https://transmathematica.org/index.php/journal/article/view/19/23 Continuum between addition, multiplication and exponentiation https://math.stackexchange.com/questions/1269643/continuum-between-addition-multiplication-and- exponentiation La quinta operación aritmética (The fifth arithmetical operation) New Numerical Methods: The Rational Mean (book) - Domingo Gomez Morin https://www.amazon.com/gp/product/1520717245/ref=dbs_a_def_rwt_hsch_vapi_tpbk_p1_i1 https://www.youtube.com/watch?v=6lORU03yuvY AULOS. LA OTRA LUZ. Music and Consonance. New musical scale not based on the Octave. https://www.youtube.com/watch?v=gbK_V_7ivDA https://domingogomezmorin.wordpress.com/ Setoid - https://en.wikipedia.org/wiki/Setoid Some ternary quasigroups over small sets http://tamivox.org/dave/math/tern_quasi/index.html The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms - Kenneth A. Ribet – https://math.berkeley.edu/~ribet/trinity.pdf Engel expansion - https://en.wikipedia.org/wiki/Engel_expansion Arithmetic Geometric Mean – https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, π, and the Ladies Diary Gert Almkvist and Bruce Berndt - https://link.springer.com/chapter/10.1007%2F978-3-319-32377-0_8 The total differential, the Cauchy-Riemann equations and the Elysian infinitesimals - Kerry Bemis Heinz mean - https://en.wikipedia.org/wiki/Heinz_mean Identric mean - https://en.wikipedia.org/wiki/Identric_mean Logarithmic mean - https://en.wikipedia.org/wiki/Logarithmic_mean Knot sum - https://mathworld.wolfram.com/KnotSum.html Hypertrascendental number - https://en.wikipedia.org/wiki/Hypertranscendental_number Infinite compositions of analytic functions https://en.wikipedia.org/wiki/Infinite_compositions_of_analytic_functions Monus - https://en.wikipedia.org/wiki/Monus Racks and quandles - https://en.wikipedia.org/wiki/Racks_and_quandles Absorption law - https://en.wikipedia.org/wiki/Absorption_law A quest for Exactness : machines, algebra and geometry for tractional constructions of differential equations -Pietro Milici – https://tel.archives-ouvertes.fr/tel-01889365/document ( See section "7.3 Open problems and perspectives" ) Multiplicative calculus - https://en.wikipedia.org/wiki/Multiplicative_calculus Subderivative https://en.wikipedia.org/wiki/Subderivative Fractal derivative - https://en.wikipedia.org/wiki/Fractal_derivative Alternative mathematical notation and its applications in calculus - Jakub Marian https://jakubmarian.com/data/bachelor_thesis.pdf Mathematics Without Numbers Towards a Modal-Structural Interpretation - Geoffrey Hellman https://www.amazon.com/Mathematics-without-Numbers-Modal-Structural-Interpretation/dp/ 0198240341 Science Without Numbers A Defense of Nominalism - Hartry
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